BROWN v. SMALLWOOD 1915
This is not a US supreme court case, it is the Minnesota Supreme Court.
We are only mentioning it here because a lot of voting activists on all sides seem to
have drawn numerous conflicting conclusions from this case.
It ruled the "Bucklin" voting system then in use in Minnesota (which also was used in
several other states, but they were not affected by this decision) unconstitutional.
Our detailed discussion.

COLEGROVE v. GREEN 1946
Court ruled that gerrymandering was political not judicial decision so
they were dodging the whole issue and (hence) all gerrymandering was
ok and not judicable at all.

BAKER v. CARR 1962
6-2 they ruled it was judicable, ha ha, we were only kidding last
time. (Even though a lower court had dismissed the case.)

Note: essentially every supreme court decision about gerrymandering from
here on seems to take it for granted that "districts" exist, which
with PR would not necessarily have been true. The supreme court
nowhere that I saw even considered the possibility of a multi-member
district and how that might affect things. However, later there will be cases specifically
about plans involving multimember districts.

REYNOLDS v. SIMS 1964, WESBERRY v. SANDERS 1964
http://caselaw.lp.findlaw.com/scripts/getcase.pl?court=US&vol=376&invol=1
Court (8-to-1 in each case) ruled some gerrymandering is beyond what is
acceptable. This was for both statehouse districts (the court
evidently feeling its power overruled whatever "state's rights" those
states had to decide their own districts) and in WvS for federal
districts within states.
Among the more egregious pre-Reynolds disparities (compiled by
Congressman Morris K. Udall, wikipedia):

In Connecticut one House district had 191 people; another, 81,000.

In New Hampshire one township with three people had a state
assemblyman; this was the same representation given another district
with 3,244. The vote of a resident of the first township might therefore
seem to be about 1000 times more powerful at the Capitol.

In Utah the smallest district had 165 people, the largest 32,380
(196 times the population of the other).

In Vermont the smallest district had 36 people, the largest
35,000, a ratio of almost 1,000 to 1.

Decision based on principle of "one man, one vote":
"Legislators represent people, not trees or acres. Legislators are
elected by voters, not farms or cities or economic interests.
And, if a State should provide that the votes of citizens in one part
of the State should be given two times, or five times, or ten times
the weight of votes of citizens in another part of the State, it could
hardly be contended that the right to vote of those residing in the
disfavored areas had not been effectively diluted. Of course, the
effect of state legislative districting schemes which give the same
number of representatives to unequal numbers of constituents is
identical. Weighting the votes of citizens differently, by any method
or means, merely because of where they happen to reside, hardly seems
justifiable"

IN OTHER WORDS: every voter should have equal power.
(This perhaps could be interpreted as an argument that IRV is
unconstitutional... because in IRV we can set up situations where my vote says A>B and
yours says B>A and my vote counts and affects a crucial AB tie, but your vote
does not count and does not affect that tie. Under the right circumstances this effect could
discriminate against one particular political type of voter. This in fact happened in
Peru 2006. But IRV's unconstitutionality
is certainly not clear though.)

"...Logically, in a society ostensibly grounded on representative
government, it would seem reasonable that a majority of the people of
a State could elect a majority of that State's legislators."

IN OTHER WORDS: the supremes endorse the "majority property" that
a majority has the power to elect a majority... but the whole
idea of having single-winner districts at all contradicts that
(since a voter majority distributed badly within
those districts would not have the power to elect a majority at all)
and we know the supremes endorse single member districts, so this is
a self-contradiction.

And it is not just a theoretical self-contradiction, it is a very real problem, because
voter-majorities (such as 60% of Ohio voters in 2006) electing small
minorities of representatives (such as 39% democratic congressmen elected by Ohio)
happen all the time as a consequence of rampant
partisan gerrymandering.
So you might imagine the Supreme Court has in REYNOLDS outlawed gerrymandering.
But that is not what the Supremes themselves imagined, since
later they explicitly legitimized partisan gerrymandering
(skip down to 2001).

"...Diluting the weight of votes because of place of residence, impairs
basic constitutional rights under the Fourteenth Amendment just as
much as invidious discriminations based upon factors such as race or
economic status."

"...We are told that the matter of apportioning representation in a
state legislature is a complex and many-faceted one. We are advised
that States can rationally consider factors other than populations. We
are admonished not to restrict the power of the States to impose
differing views as to political philosophy on their citizens. We are
cautioned about the dangers of entering into political thickets and
mathematical quagmires. Our answer is this: a denial of
constitutionally protected rights demands judicial protection; our
oath and our office require no less of us. To the extent that a
citizen's right to vote is debased, he is that much less a citizen.
The weight of a citizen's vote cannot be made to depend on where he
lives. Population is, of necessity, the starting point for
consideration and the controlling criterion for judgement in
legislative apportionment controversies. A citizen, a qualified voter,
is no more nor no less so because he lives in the city or on the farm.
This is the clear and strong command of our Constitution's Equal
Protection Clause. This is at the heart of Lincoln's vision of
"government of the people, by the people,...for the people." We hold
that, as a basic constitutional standard, the Equal Protection Clause
requires that the seats in both houses of a bicameral state
legislature must be apportioned on a population basis. Simply stated,
an individual's right to vote for state legislators is
unconstitutionally impaired when its weight is in a substantial
fashion diluted when compared with votes of citizens living in other
parts of the state."

IN OTHER WORDS: All voters must have equal power, but the court
intentionally refuses to provide a precise definition of exactly
what that means.

FORTSON v. DORSEY 1965
http://www.vlex.us/caselaw/U-S-Supreme-Court/Fortson-v-Dorsey-379-U-S-433-1965/2100-19993246%2C01.html
This case seems to come the closest so far to addressing the question of
whether multi-member districts and PR are constitutionally
permissible.
In Georgia some counties were single-member districts,
but other counties (the 7 most populous ones) had to elect more
than one member to keep the 1-man-1-vote principle approximately
satisfied, so they made those counties be multi-member districts
in which the whole coterie of members got elected in one county-wide
vote. The lower court held this scheme was discriminatory.
The supremes reversed that, holding it was not necessary
for all districts to be single-member, to satisfy the
US constitution's equal protection clause.
They noted that in REYNOLD v. SIMS they had already said
"One body could be composed of single-member districts while the
other could have at least some multi-member districts."
They in FORTSON noted that
"the appellees assert that this scheme is defective because
county-wide voting in multi-district counties could, as a matter of
mathematics, result in the nullification of the unanimous choice of
the voters of a district, thereby thrusting upon them a senator for
whom no one in the district had voted. But this is only a highly
hypothetical assertion... [and]
It is not accurate to treat a senator from a multi-district county as
the representative of only that district within the county wherein he
resides...
If the weight of the vote of any voter in a Fulton County district,
when he votes for seven senators to represent him in the Georgia
Senate, is not the exact equivalent of that of a resident of a
single-member constituency, we cannot say that his vote is not
"approximately equal in weight to that of any other citizen in the State."

"...our opinion is not to be understood to say that in all instances
or under all circumstances such a system as Georgia has will comport
with the dictates of the Equal Protection Clause. It might well be
that, designedly or otherwise, a multi-member constituency
apportionment scheme, under the circumstances of a particular case,
would operate to minimize or cancel out the voting strength of racial
or political elements of the voting population...
This question, however, is not presented by the record before us
[or if it is, only in a too-trivial, not analysed-enough way]."

CONCLUSIONS: Does this mean that the court believes PR
is constitutional? Definitely yes for statehouse races
for some – but they caution, perhaps not all – multimember
election methods. Case does not address the question of
federal races.

For example, it might well be speculated based on this decision that
the court would say that party-list PR is unconstitutional in
a situation where the parties had the wrong properties –
or that a multimember district elected via first-past-the-post
with 49% blacks is not constitutional whereas a natural split
of that district into some single-member
districts some of which had black majorities would
be constitutional. It is not clear. But my top guess would
be that the court would rule that any PR scheme which
was not fairly obviously designed to be discriminatory, would be ok.

The court also made clear here its distaste for hypothetical mathematical scenarios –
they insist on real historical examples of the phenomenon.

Another comment (to clarify how it could be possible to abuse multi- and single-member
districting): The party in power could create lots of single-districts
where it has 50-60% support and then
(say) 2-seaters to prevent the minority getting much or any representation.
[However, if they can gerrymander that effectively, they would probably already
be massively over-represented.]

Finally note that here the Court says that design, i.e. intent, is not necessary, merely
effect is necessary, to make a discriminatory scheme illegal.

CONGRESS IN 1967
passed a law (Public Law 90-196 Title 2 U.S.C. §2c) saying
in each state entitled to more than one rep
under the decennial census
"there shall be established by law a number of districts equal to the
number of Representatives to which such State is so entitled, and
Representatives shall be elected only from districts so established,
no district to elect more than one Representative."

This law outlaws PR in federal elections, and remains in force today (2006).
However, a law is not the same thing
as the constitution, so if this law is got rid of, then
perhaps PR would be permitted.

IN THE 1970s
the Supreme Court made clear its preference for
single-member legislative districts by discouraging the use of
multimember districts in court-drawn plans.

CONNOR v. JOHNSON 1971
Court held that, as a general rule, single-member districts are
preferable to large multimember districts when district courts are
required to fashion apportionment plans.

CHAPMAN v. MEIER 1975
Court stated that "(t)he standards for evaluating the use of multimember
districts thus clearly differ depending on whether a federal court or
a state legislature has initiated the use ...
Absent particularly pressing features calling for multimember
districts, a United States district court should refrain from imposing
them upon a State."

WHITCOMB v. CHAVIS 1971
multimember districts violate the Equal Protection Clause if the
districts were "conceived or operated as purposeful devices to further
racial or economic discrimination."

(So it isn't just race. It also could be economics, and in other cases the court said it
also could be geography or political bent.)

WHITE v. REGESTER 1973
Court invalidated the use of multimember legislative districts in two
Texas counties because the redistricting plan had operated to cancel
out or minimize the voting strength of Black and Mexican American
communities.

"Plainly, under our cases, multimember districts are not per se
unconstitutional, nor are they necessarily unconstitutional when used
in combination with single-member districts in other parts of the
State. But we have entertained claims that multimember districts are
being used invidiously to cancel out or minimize the voting strength
of racial groups. To sustain such claims, it is not enough that the
racial group allegedly discriminated against has not had legislative
seats in proportion to its voting potential. The plaintiffs' burden is
to produce evidence to support findings that the political processes
leading to nomination and election were not equally open to
participation by the group in question other residents in the district to participate in
the political processes and to elect legislators of their choice.

COMMENT:
This is very imprecise. At first it
sounds like PR schemes all are ok...

HARPER v. VIRGINIA BOARD OF ELECTIONS 1966
Court overruled its decision in Breedlove v. Suttles 1937 in view of the 24th amendment
(ratified 1964), now declaring by 8-1 vote that
all poll taxes are illegal including in state elections
(even though the text of the amendment does not explicitly discuss non-federal elections,
they deduced this using 14th amendment).
In a two-month period in early 1966, Federal courts declared unconstitutional poll
tax laws in the last four states to have them: Texas (9 February),
Alabama (3 March), and Virginia (25 March).
Mississippi's $2.00 poll tax was the last to fall, declared
unconstitutional on 8 April 1966, by a federal panel in Jackson, Mississippi.
Virginia tried to dodge by allowing for the poll tax to be waived if the
would-be voter filed a certificate of residency six months prior to the election
(onerous paperwork). The supremes (unanimously)
ruled even that weakened poll tax unconstitutional in
HARMAN v. FORSSENIUS 1965.

THE TEXT of the 24th amendment:

The right of citizens of the United States to vote in any primary or other election for President or Vice President, for electors for President or Vice President, or for Senator or Representative in Congress, shall not be denied or abridged by the United States or any State by reason of failure to pay any poll tax or other tax.

The Congress shall have power to enforce this article by appropriate legislation.

Various "pay to vote" schemes have been advanced by economists as hopefully superior
to traditional moneyless voting schemes.
However, I believe they all would be unconstitutional for use in the USA
until/unless (24+14)th amendment-combination is supplanted or repealed.

WHITE v. WEISER 1973
Court invalidated a Texas redistricting scheme because the maximum population deviation
was 2.4%, which they considered too large.

I would hold, as have a long line of decisions in this Court,
that Congress has ultimate supervisory power over congressional
elections. Similarly, it is the prerogative of Congress to
oversee the conduct of presidential and vice-presidential
elections and to set the qualifications for voting for electors
for those offices. It cannot be seriously contended that
Congress has less power over the conduct of presidential
elections than it has over congressional elections.

MOBILE v. BOLDEN 1980
By a 6-3 decision, court ruled that a racial gerrymander of Mobile
was ok in the sense that to kill it you had not only to prove racial
discrimination effect (which was proven, said courts)
but also intent. This would seem to
be a vast weakening of GOMILLION, FORTSON, and GAFFNEY.

THORNBURG v. GINGLES 1982
Although in MOBILE the court had required demonstration of racial
discrimination intent, they now say only effect and not intent
needs to be demonstrated, which change they justified because
congress in the meantime had passed new amendments to the voting
rights act.

Court held that, unless there is a conjunction of the following
circumstances, the use of multimember legislative districts generally
will not impede the ability of minority voters to elect
representatives of their choice:

The minority group must demonstrate that it is sufficiently large
and geographically compact to constitute a majority in a single-member
legislative district;

The minority group must show that it is politically cohesive; and

The minority group must demonstrate that the majority votes
sufficiently as a bloc to enable the majority to usually defeat the
preferred candidate of the minority.

COMMENT:
Whoa!! This definitely seems to make PR
unconstitutional at least under the right (artifically constructable)
circumstances (which we've discussed already).

A districting plan for Indiana, which actually involved some "PR"
multimember districts, was held to be invalid (due to it being
gerrymandering) by district court, which found
numerous "bright lines" indicating an anti-Democratic gerrymander
including contorted shapes of many districts, the inconsistent
use of multi-member districts, and the "unashamedly partisan" motivation for the districting plan.
(But unequal district populations were not an issue – maximum deviation<1.2%.)
But the US supreme court by a 6-3 vote
reversed the district court. Nevertheless the US supreme court
did hold that in principle, gerrymandering was outlawed (including
of precisely this sort)
and that it was decidable by courts whether it had occurred.
They just claimed the standard of proof required, was beyond that
actually achieved in this case.

O'Connor, Rehnquist, & Chief Justice Burger all claimed gerrymandering
was subjective and hence not even decidable by courts at all.
Except if it were race based. Then it was decidable and objective.
But they were in the minority so nobody cared.
(O'Connor also expressed the remarkable opinions that most gerrymandering was bipartisan,
so neither party is really hurt, and hence there is nobody hurt so no basis to sue –
the idea that voters might thereby be denied all control, contrary to
the ideals of all democracy, did not having appear to have penetrated her skull –
plus supposedly gerrymandering is self-limiting since overreaching can backfire for a party;
plus it is just a spoil of victory for the winning party, hence fine and not to be denied
by any judge. I find the level of her ignorance appalling.)

(Incidentally, a lot of town boundaries and county boundaries are race-based.
A fact none of them mentioned.)

Bandemer created a standard for finding an unacceptable partisan gerrymander
that the
plurality admitted was "difficult of application." That difficulty resulted in
relatively little
litigation in this area during the 1990s that shed new light on the issue. The
standard
created in Bandemer requires proof of both discriminatory effect and intent,
which seems once again to reverse THORNBURG which had reversed GOMILLION.

MILLER v. JOHNSON 1995
and
SHAW v. RENO 1993
Majority-black districts in Georgia and North Carolina are
ruled unconstitutional. These decisions appear to contradict
UNITED JEWISH ORG v CAREY 1977 (which indeed had been the basis for lower
courts and agencies ruling that these majority-black districts were necessary –
indeed in NC just one of them was not enough) but supposedly for subtle and mysterious reasons,
they don't really contradict.

Michael W. McConnell
(the darling judge of conservatives and thought to be a prime candidate for Bush's next
supreme court appointment) in an article in
Harvard J. Law & Public Policy 24,1 (Fall 2000),
argued that the whole 1-man-1-vote principle embarked on in REYNOLDS v. SIMS
was a mistake and really, back in the good old days when districts were drawn based on
traditional city lines, geographical features, etc, gerrymandering was prevented,
but now, with 1-man-1-vote forcing all districts to be equal in population,
those old traditions had to be abandoned and the result was totally opening
the door to massive gerrymandering. (?!) McConnell says KARCHER v. DAGGETT 1983 was the ultimate
example of idiocy;
he claims in this case the Supremes invalidated some New Jersey districting having
a maximum population deviation of 0.698% as "too unequal," which McConnell
claims was below statistical error anyhow. (A Pennsylvania districting
with maximum deviation of 0.4% was allowed by a different federal court
535 F.Supp. 191, M.D. Penn. 1982, and much looser
standards have been applied to state-house
districtings, e.g.
lower courts sanctioned population deviations of nearly 14% in New Hampshire
and 11% in Montana [Bayer v. Gardner, 540 F.Supp. 624 (N.H. 1982);
McBride v. Mahoney, 573 F.Supp. 913 (Mont. 1983)] while
a successful challenge was made to a Virginia redistricting plan containing
a deviation of 26.63% [Cosner v. Dalton, 522 F. Supp. 350 (E.D. Va. 1981)].)
Meanwhile, McConnell claims,
some districts can have very unequal voting populations despite equal populations;
he claims to have an district-pair example where it's 282,000 versus 429,000 voting pop.
(Recently that has mattered particularly in upstate New York districts
containing large prisons whose
inmates can't vote but count as "population.")

Justice J.P.Stevens in KARCHER outlined his view of proposed standards for
invalidating a districting statute on grounds of partisan gerrymandering.
He claimed "political gerrymandering is one species of 'vote dilution' that is
proscribed by the Equal Protection Clause," Stevens
clearly delineated the shortcomings of analyses based purely on population criteria.
However, unfortunately, Stevens'
view appears never to have swayed enough other justices to carry legal force,
indeed when the district court in BANDEMER attempted to use Stevens' guidelines it was
overruled.
To quote a review of this by C.Backstrom & L.Robins:

Stevens proposed that prima facie evidence of political gerrymandering could be found in
"drastic departures from compactness,"
"extensive deviation from established political boundaries,"
or a procedural history that "excluded divergent viewpoints, openly reflected the use of
partisan criteria, and provided no explanation of the reasons for selecting one plan over another."
To Stevens such considerations would not be full proof but mere
evidence of unconstitutional gerrymandering, a rebuttable presumption that the state could
overcome by demonstrating that "the plan as a whole embodies acceptable, neutral objectives."
Applying his standards to the New Jersey plan, Stevens found unjustifiable violations of
compactness, a decision-making process that was "far from neutral," and no legitimate
justifications for the irregularities of the plan advanced by the state.

ABRAMS v. MILLER 1997
all but required that states follow vague traditional principles such as
contiguity, compactness, respect for political boundaries, and population equality;
it upheld districts districts drawn to replace those invalidated by MILLER
because they followed county lines a lot, etc.

RENO v. BOSSIER PARISH SCHOOL BD. 2000
By 5-4, Supreme court said the Justice Department's approval for redistricting (under
voting rights act) cannot
be withheld when the new redistricting plan was adopted with a discriminatory
purpose, unless it worsens matters for minority voters.

COMMENT:
This contradicts previous cases
like CHAPMAN v. MEIER where the Supremes had ruled it was not just race.
Now, it is just racial discrimination that is bad;
other kinds of discriminatory districting are fine.

HUNT v. CROMARTIE and SMALLWOOD v. CROMARTIE 2001
By 5-4, Supreme court upheld the much-litigated North
Carolina 12th congressional district, saying those who challenged it did
not show that race was the main factor in its creation.

This was the fourth time that the high court had looked at that
district. In the meantime there'd been a new census so the Supremes' decision
was kind of moot on arrival. The case is a follow-up to SHAW v. RENO 1993 ruling
that racially drawn districts may violate the rights of white voters.
"The evidence ... does not show that racial considerations predominated in
the drawing of District 12's boundaries," Justice Stephen G. Breyer wrote
for the court. "That is because race in this case correlates closely with
political behavior."
Those who challenged the district have "not successfully shown that race,
rather than politics, predominantly accounts for the result."

COMMENT:
These two cases completely reverse and ignore
the statement in FORTSON that redistricting could be bad if
it "would operate to minimize or cancel out the voting strength of racial
or political elements of the voting population"
and now says it has to be predominantly race to be bad. This ludicrous
decision in my view legitimizes
race-discriminatory gerrymandering in all the cases where it has a political effect, i.e.
all the cases where anybody would want to do it?!?!
I mean, nowadays, the best way to gerrymander is using computers based on political results,
and the computers know nothing about racial composition, so Q.E.D, all gerrymandering is fine.
These rulings also explicitly legitimize gerrymandering for political purposes.
BUSH v. VERA 1996 also explicitly legitimizes political gerrymandering (of Texas by the Democrats)
on a massive and blatant scale.

On the other hand in SHAW II (delivered on the same day as BUSH) and BUSH v. VERA the court
noted that violations of traditional vague districting principles
cause symbolic and "expressive" harm
by sending "pernicious messages" to voters and politicians.
But that was just guff. The solid stuff is the stuff legitimizing gerrymandering big time.
E.g. Rehnquist in SHAW II explicitly denies a claim in Stevens' dissent:

"In his dissent, Justice Stevens argues that strict scrutiny does not apply where a State
respects or compl(ies) with traditional districting principles ...
That, however, is not the standard announced and applied in MILLER,
where we held that strict scrutiny applies when race is the
predominant
consideration in drawing the district lines."

DECKER v. BRADBURY 2002
Supreme Court refused to rule on Oregon's unique vote-by-mail program, which
eliminated most polling places. The 9th circuit court ruling
on the case (permitting the system) therefore stood:
since Oregon did not count the mail-in votes until
election day, the voting was not improper under federal laws.
Oregon is the only state that has ended precinct polling but more
than half the voters in Washington State also voted by mail.

GRUTTER v. BOLLINGER
and GRATZ v. BOLLINGER 2003
in these cases, white students rejected by University of Michigan
sued trying to overturn U.M.'s "affirmative action" admissions policies.
The supremes split the difference by ruling
that U.M.'s policy was fine for its law school (GRUTTER)
but unconstitutional for its undergraduate admissions (GRATZ).
Although these cases were not about elections per se,
these affirmative action policies can be thought of as "proportional representation"
and the court here is ruling in GRATZ that such policies are in general unconstitutional, although
in GRUTTER they are sometimes permissible if the need is arguably large enough and the
policy is "narrowly tailored" enough (whatever that means).

COMMENT:
Fiji and New Zealand both have race-based quotas in their
parliaments, e.g. a certain number of seats are set aside for MPs of Maori race, etc.
The Socialist Party of the USA similarly has gender-based quotas in internal elections,
causing 50-50 gender splits.
These all can be thought of as a form of "proportional representation."
Would these schemes be legal in USA governments? (And note, US federal government
has historically been extremely male-dominated.)
The Supreme Court apparently has never given us any direct
guidance on that question. University of Michigan was a state institution, and
if the obvious parallel is drawn we would conclude that Fiji-like or Socialist-like PR
quota systems are unconstitutional – except when they are constitutional, which
in these cases seems basically to be a matter of opinion. I could based on these cases
argue it either way.

OVERALL CONCLUSION (BY ME):
The supremes constantly contradict and overrule themselves in an apparently random manner,
often displaying an amazing ignorance of reality and disregard for
principles of democracy as they do so. There are certainly some
themes you can see,
but there often seems little basis for making any confident prediction of what they
will rule next.